On-chip thermometry for microwave optomechanics implemented in a nuclear demagnetization cryostat

We report on microwave optomechanics measurements performed on a nuclear adiabatic demagnetization cryostat, whose temperature is determined by accurate thermometry from below 500$~\mu$K to about 1$~$Kelvin. We describe a method for accessing the on-chip temperature, building on the blue-detuned parametric instability and a standard microwave setup. The capabilities and sensitivity of both the experimental arrangement and the developed technique are demonstrated with a very weakly coupled silicon-nitride doubly-clamped beam mode of about 4$~$MHz and a niobium on-chip cavity resonating around 6$~$GHz. We report on an unstable intrinsic driving force in the coupled microwave-mechanical system acting on the mechanics that appears below typically 100$~$mK. The origin of this phenomenon remains unknown, and deserves theoretical input. It prevents us from performing reliable experiments below typically 10-30$~$mK; however no evidence of thermal decoupling is observed, and we propose that the same features should be present in all devices sharing the microwave technology, at different levels of strengths. We further demonstrate empirically how most of the unstable feature can be annihilated, and speculate how the mechanism could be linked to atomic-scale two level systems. The described microwave/microkelvin facility is part of the EMP platform, and shall be used for further experiments within and below the millikelvin range.